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arxiv: 2507.20247 · v2 · submitted 2025-07-27 · 🧮 math.GT · math.DG

Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds

Ervin Hadziosmanovic This is my paper

Pith reviewed 2026-05-19 02:37 UTC · model grok-4.3

classification 🧮 math.GT math.DG
keywords negatively curved manifoldsbounded fundamental classCheeger isoperimetric constantinfinite volumebounded geometryisoperimetric inequalityvolume class vanishing
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The pith

For negatively curved manifolds of infinite volume with curvature bounded away from zero and bounded geometry, the bounded fundamental class vanishes if and only if the Cheeger isoperimetric constant is positive.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an equivalence between the vanishing of the bounded fundamental class and the positivity of the Cheeger isoperimetric constant for manifolds with negative curvature bounded away from zero, infinite volume, and bounded geometry. The bounded fundamental class arises from integrating the volume form over straight top-dimensional simplices. One direction of this equivalence holds more generally, without the bounded geometry assumption, showing that a positive Cheeger constant forces the class to vanish. A sympathetic reader would care because this links a bounded-cohomology invariant directly to a geometric isoperimetric quantity in infinite-volume settings.

Core claim

For manifolds with negative curvature bounded away from zero, of infinite volume and with bounded geometry, the bounded fundamental class vanishes if and only if the Cheeger isoperimetric constant is positive. The bounded fundamental class is defined via integration of the volume form over straight top-dimensional simplices. Without the bounded geometry assumption, positivity of the Cheeger constant still implies vanishing of the bounded volume class.

What carries the argument

The bounded fundamental class, defined by integrating the Riemannian volume form over straight top-dimensional simplices.

If this is right

  • The equivalence between vanishing of the bounded fundamental class and positivity of the Cheeger constant holds when bounded geometry is assumed.
  • Positivity of the Cheeger constant implies vanishing of the bounded volume class for all such manifolds even without bounded geometry.
  • The two invariants detect the same kind of largeness in these negatively curved infinite-volume manifolds under the stated conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result may suggest similar equivalences for other curvature bounds or different constructions of bounded classes in geometric topology.
  • It could lead to new ways of computing or bounding the Cheeger constant using simplicial volume techniques on explicit examples such as hyperbolic manifolds with funnels.
  • One might test whether the equivalence persists when bounded geometry is weakened to other regularity conditions.

Load-bearing premise

The manifolds possess bounded geometry, which is required to obtain the equivalence in both directions.

What would settle it

A manifold with negative curvature bounded away from zero, infinite volume, and bounded geometry, in which the bounded fundamental class vanishes but the Cheeger constant is zero (or the Cheeger constant is positive but the class does not vanish).

read the original abstract

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes if and only if the Cheeger isoperimetric constant is positive. This gives a partial affirmative answer to a conjecture of Kim and Kim. Furthermore, we show that for all manifolds with negative curvature bounded away from $0$ of infinite volume, the positivity of the Cheeger constant implies the vanishing of the bounded volume class, solving one direction of the conjecture in full generality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper proves that for manifolds with sectional curvature bounded above by a negative constant, infinite volume, and bounded geometry, the bounded fundamental class (defined via integration of the volume form over straight geodesic top-dimensional simplices) vanishes if and only if the Cheeger isoperimetric constant is positive. It further shows that positivity of the Cheeger constant implies vanishing of the bounded volume class for all such manifolds of infinite volume without requiring bounded geometry, thereby giving a partial affirmative answer to a conjecture of Kim and Kim.

Significance. If the arguments hold, the result provides a geometric characterization linking bounded cohomology invariants to isoperimetric properties in non-compact negatively curved manifolds. The clean separation of the two directions—one holding in full generality and the other under the bounded-geometry hypothesis—is a notable strength, as is the use of the standard straight-simplex definition of the bounded fundamental class together with the curvature lower bound to guarantee uniform boundedness of the cochain.

major comments (2)
  1. [§4, Theorem 4.1] §4, Theorem 4.1 (converse direction): the argument that bounded geometry implies the required uniform control on simplex volumes and hence the vanishing of the class when the Cheeger constant is positive appears to rely on a comparison with the model space of constant curvature −c; an explicit reference or short estimate showing how the bounded-geometry hypothesis produces the necessary lower bound on injectivity radius or volume growth would strengthen the exposition.
  2. [§3.3, Proposition 3.8] §3.3, Proposition 3.8: the reduction from the Cheeger-constant positivity to the vanishing of the bounded class uses a filling argument; it is not immediately clear whether the constants in the filling inequality depend on the curvature bound in a way that remains uniform when the manifold is non-compact, and a short remark addressing this uniformity would be helpful.
minor comments (3)
  1. [Introduction] In the introduction, the statement of the main theorem could explicitly list the curvature hypothesis (sec ≤ −c < 0) alongside the bounded-geometry assumption to avoid any ambiguity about which hypotheses apply to each direction.
  2. [§2 and §4] Notation for the bounded fundamental class [M]_b and the straight-simplex cochain is introduced in §2 but used without repeated reminder in later sections; a brief parenthetical recall of the definition in the statement of Theorem 4.1 would improve readability.
  3. [References] The reference list contains the Kim–Kim conjecture paper but omits a citation to the original definition of the bounded fundamental class via straight simplices (e.g., the work of Gromov or subsequent papers using this construction); adding one or two such references would place the construction in context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address the major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4, Theorem 4.1] §4, Theorem 4.1 (converse direction): the argument that bounded geometry implies the required uniform control on simplex volumes and hence the vanishing of the class when the Cheeger constant is positive appears to rely on a comparison with the model space of constant curvature −c; an explicit reference or short estimate showing how the bounded-geometry hypothesis produces the necessary lower bound on injectivity radius or volume growth would strengthen the exposition.

    Authors: We agree that an explicit reference or short estimate would improve clarity. In the revised version we will insert a brief paragraph immediately following the statement of Theorem 4.1. This paragraph will recall the standard volume comparison (via the Rauch comparison theorem) between the given manifold and the model space of constant curvature −c, and will note that the bounded-geometry hypothesis supplies a uniform positive lower bound on the injectivity radius, which in turn yields a uniform upper bound on the volumes of straight simplices independent of basepoint. revision: yes

  2. Referee: [§3.3, Proposition 3.8] §3.3, Proposition 3.8: the reduction from the Cheeger-constant positivity to the vanishing of the bounded class uses a filling argument; it is not immediately clear whether the constants in the filling inequality depend on the curvature bound in a way that remains uniform when the manifold is non-compact, and a short remark addressing this uniformity would be helpful.

    Authors: We thank the referee for this observation. The constants appearing in the filling inequality depend only on the dimension and the fixed upper curvature bound −c < 0; the argument itself is local and does not rely on global compactness. We will add a short clarifying remark at the end of the proof of Proposition 3.8 stating that these constants remain uniform for non-compact manifolds under the given curvature hypothesis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an if-and-only-if equivalence between the vanishing of the bounded fundamental class (defined via integration of the volume form against straight geodesic simplices, a standard construction in bounded cohomology) and positivity of the Cheeger isoperimetric constant. One direction holds under the stated curvature and infinite-volume hypotheses alone; the converse invokes bounded geometry as an explicit additional hypothesis. No parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the logical steps rely on independent geometric estimates rather than reducing to the input definitions by construction. The result is therefore a genuine theorem rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Riemannian geometry assumptions and the specific definitions of the bounded class and Cheeger constant; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • domain assumption Manifolds are complete Riemannian manifolds with sectional curvature bounded above by a negative constant.
    Explicitly stated in the abstract as the curvature condition for the manifolds under consideration.
  • domain assumption The bounded fundamental class is defined by integrating the volume form over straight top-dimensional simplices.
    Definition supplied directly in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Bounded cohomology classes from differential forms

    math.GT 2026-04 unverdicted novelty 6.0

    Integration of closed bounded 2-forms over geodesic simplices gives an injective map from the space of such forms into H^2_b(M) for complete hyperbolic n-manifolds whose fundamental group has limit set equal to the fu...

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Tameness of hyperbolic 3-manifolds

    Ian Agol. Tameness of hyperbolic 3-manifolds. arXiv preprint math/0405568, 2004. doi: 10.48550/arXiv.math/0405568

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.math/0405568 2004

  2. [2]

    Quasi-isometry classification of some manifolds of bounded geometry.Mathe- matische Zeitschrift, 217:501–528, 1994.doi:10.1007/BF02572337

    Oliver Attie. Quasi-isometry classification of some manifolds of bounded geometry.Mathe- matische Zeitschrift, 217:501–528, 1994.doi:10.1007/BF02572337

    work page doi:10.1007/bf02572337 1994

  3. [3]

    Aperiodic tilings, positive scalar curvature, and amenability of spaces.Journal of the American Mathematical Society, 5(4):907–918, 1992

    Jonathan Block and Shmuel Weinberger. Aperiodic tilings, positive scalar curvature, and amenability of spaces.Journal of the American Mathematical Society, 5(4):907–918, 1992. doi:10.2307/2152713

    work page doi:10.2307/2152713 1992

  4. [4]

    Bouts des variétés hyperboliques de dimension 3.Annals of Mathematics, 124(1):71–158, 1986.doi:10.2307/1971388

    Francis Bonahon. Bouts des variétés hyperboliques de dimension 3.Annals of Mathematics, 124(1):71–158, 1986.doi:10.2307/1971388

    work page doi:10.2307/1971388 1986

  5. [5]

    Bowditch

    Brian H. Bowditch. Geometrical finiteness with variable negative curvature.Duke Mathemat- ical Journal, 77(1):229–274, 1995.doi:10.1215/S0012-7094-95-07709-6

    work page doi:10.1215/s0012-7094-95-07709-6 1995

  6. [6]

    Some Riemannian and dynamical invariants of foliations

    Robert Brooks. Some Riemannian and dynamical invariants of foliations. InDifferential ge- ometry, volume 32 ofProgress in Mathematics, pages 56–72, 1983

    work page 1983

  7. [8]

    Shrinkwrapping and the taming of hyperbolic 3- manifolds

    Danny Calegari and David Gabai. Shrinkwrapping and the taming of hyperbolic 3- manifolds. Journal of the American Mathematical Society, 19(2):385–446, 2006.doi:10. 1090/S0894-0347-05-00513-8

    work page 2006

  8. [9]

    Richard D. Canary. Ends of hyperbolic 3-manifolds.Journal of the American Mathematical Society, 6(1):1–35, 1993.doi:10.2307/2152793

    work page doi:10.2307/2152793 1993

  9. [10]

    Riemannian geometry: a modern introduction.Number98inCambridgestudies in advanced mathematics

    IsaacChavel. Riemannian geometry: a modern introduction.Number98inCambridgestudies in advanced mathematics. Cambridge University Press, 2nd edition, 2006

    work page 2006

  10. [11]

    Bounded Cohomology of Discrete Groups, volume 227 of Mathematical Surveys and Monographs

    Roberto Frigerio. Bounded Cohomology of Discrete Groups, volume 227 of Mathematical Surveys and Monographs. American Mathematical Society, 2017.doi:10.1090/surv/227

    work page doi:10.1090/surv/227 2017

  11. [12]

    Volume and bounded cohomology.Publications Mathématiques de l’IHÉS, 56:5–99, 1982

    Michael Gromov. Volume and bounded cohomology.Publications Mathématiques de l’IHÉS, 56:5–99, 1982

    work page 1982

  12. [13]

    Hyperbolic manifolds, groups and actions

    Mikhael Gromov. Hyperbolic manifolds, groups and actions. InRiemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, volume 97 ofAnnals of mathematics studies, pages 183–213. Princeton University Press, 1981

    work page 1978

  13. [14]

    Small eigenvalues of geometrically finite manifolds.The Journal of Geo- metric Analysis, 14:281–290, 2004.doi:10.1007/BF02922073

    Ursula Hamenstädt. Small eigenvalues of geometrically finite manifolds.The Journal of Geo- metric Analysis, 14:281–290, 2004.doi:10.1007/BF02922073

    work page doi:10.1007/bf02922073 2004

  14. [15]

    Topology, 21(1):83–89, 1982.doi:10.1016/0040-9383(82)90043-X

    HisaoInoueandKoichiYano.TheGromovinvariantofnegativelycurvedmanifolds. Topology, 21(1):83–89, 1982.doi:10.1016/0040-9383(82)90043-X

    work page doi:10.1016/0040-9383(82)90043-x 1982

  15. [16]

    Johnson.Cohomology in Banach algebras, volume 127 ofMemoirs of the American Mathematical Society

    Barry E. Johnson.Cohomology in Banach algebras, volume 127 ofMemoirs of the American Mathematical Society. American Mathematical Society, 1972

    work page 1972

  16. [17]

    Bounded cohomology and the Cheeger isoperimetric con- stant

    Sungwoon Kim and Inkang Kim. Bounded cohomology and the Cheeger isoperimetric con- stant. Geometriae Dedicata, 179(1):1–20, 2015.doi:10.1007/s10711-015-0064-x

    work page doi:10.1007/s10711-015-0064-x 2015

  17. [18]

    Heat semigroup and functions of bounded variation on Riemannian manifolds

    Michele Miranda Jr., Diego Pallara, Fabio Paronetto, and Marc Preunkert. Heat semigroup and functions of bounded variation on Riemannian manifolds. Journal für die reine und angewandte Mathematik, 613:99–119, 2007.doi:10.1515/CRELLE.2007.093

    work page doi:10.1515/crelle.2007.093 2007

  18. [19]

    McGraw-Hill, Inc., 1987

    Walter Rudin.Real and complex analysis. McGraw-Hill, Inc., 1987

    work page 1987

  19. [20]

    Growth of a primitive of a differential form.Bulletin de la Société Mathématique de France, 129(2):159–168, 2001.doi:0.24033/bsmf.2390

    Jean-Claude Sikorav. Growth of a primitive of a differential form.Bulletin de la Société Mathématique de France, 129(2):159–168, 2001.doi:0.24033/bsmf.2390

    work page 2001

  20. [21]

    Undergraduate Texts in Mathematics

    Isadore Manuel Singer and John A Thorpe.Lecture notes on elementary topology and geome- try. Undergraduate Texts in Mathematics. Springer, 1976.doi:10.1007/978-1-4615-7347-0

    work page doi:10.1007/978-1-4615-7347-0 1976

  21. [22]

    Bounded cohomology and topologically tame Kleinian groups.Duke Math

    Teruhiko Soma. Bounded cohomology and topologically tame Kleinian groups.Duke Math. J., 90(1):357–370, 1997.doi:10.1215/S0012-7094-97-08814-1

    work page doi:10.1215/s0012-7094-97-08814-1 1997

  22. [23]

    Cycles for the dynamical study of foliated manifolds and complex manifolds

    Dennis Sullivan. Cycles for the dynamical study of foliated manifolds and complex manifolds. Inventiones mathematicae, 36(1):225–255, 1976.doi:10.1007/BF01390011

    work page doi:10.1007/bf01390011 1976